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The Goldbach Conjecture is a very simple conjecture to understand providing you have some basic knowledge about primes. It is one of those conjectures which is very easy to comprehend, but fiendishly difficult to solve. It was first conceived during a letter written by Goldbach to Euler in 1742. There is currently a strong and weak version of the Goldbach Conjecture, with the weak conjecture actually being the original conjecture.
The original conjecture asks to prove that all positive integers greater than 2 are equivalent to the sum of three primes, the only problem with the original conjecture, is that Goldbach considered the unit “1” to be prime, which has now been dismissed in Mathematics. However, the conjecture has been restated by Euler to the following modern equivalent, which is known as the strong Goldbach Conjecture (sometimes referred to as the binary Goldbach Conjecture):
Strong Goldbach Conjecture: Every positive even integer greater than or equal to 4, can be expressed as the sum of two primes.
The conjecture can also be expressed using the totient function:
where m is a positive integer and p and q are both prime numbers. The totient function can be defined as the following (p is prime):
The weaker version of the Goldbach conjecture asks a slightly different question, and is as follows:
Weak Goldbach Conjecture: Every odd integer greater than or equal to 9, can be expressed as the sum of three odd primes.
Similarly, Levy’s Conjecture claims that all odd integers greater than or equal to 7, can be expressed the sum of prime and twice a prime.
where n is a positive integer greater than or equal to 7, and p and q are both primes.
Goldbach Number:
Goldbach numbers are any positive even integers which are the sum of two odd primes.
Goldbach Partition:
The Goldbach Partition is the number of ways in which a positive even integer n, can be written as the sum of two primes. The function which produces these partitions is called the Goldbach Partition function
. It is defined in the following manner:
Methods for proving the Goldbach Conjecture:
The two most common methods which have been used for solving the Goldbach Conjecture involve the Hardy-Littlewood Method along with a number of different sieving methods which have been used to produce greater asymptotic bounds on the distribution of which natural numbers can be written as Goldbach Partitions.
The original conjecture asks to prove that all positive integers greater than 2 are equivalent to the sum of three primes, the only problem with the original conjecture, is that Goldbach considered the unit “1” to be prime, which has now been dismissed in Mathematics. However, the conjecture has been restated by Euler to the following modern equivalent, which is known as the strong Goldbach Conjecture (sometimes referred to as the binary Goldbach Conjecture):
Strong Goldbach Conjecture: Every positive even integer greater than or equal to 4, can be expressed as the sum of two primes.
The conjecture can also be expressed using the totient function:
where m is a positive integer and p and q are both prime numbers. The totient function can be defined as the following (p is prime):
The weaker version of the Goldbach conjecture asks a slightly different question, and is as follows:
Weak Goldbach Conjecture: Every odd integer greater than or equal to 9, can be expressed as the sum of three odd primes.
Similarly, Levy’s Conjecture claims that all odd integers greater than or equal to 7, can be expressed the sum of prime and twice a prime.
Goldbach Number:
Goldbach numbers are any positive even integers which are the sum of two odd primes.
Goldbach Partition:
The Goldbach Partition is the number of ways in which a positive even integer n, can be written as the sum of two primes. The function which produces these partitions is called the Goldbach Partition function
G(n) = #{(p, q) | n = p + q, p \le q}Methods for proving the Goldbach Conjecture:
The two most common methods which have been used for solving the Goldbach Conjecture involve the Hardy-Littlewood Method along with a number of different sieving methods which have been used to produce greater asymptotic bounds on the distribution of which natural numbers can be written as Goldbach Partitions.
Goldbach Conjecture -- from Wolfram MathWorld
Goldbach's original conjecture (sometimes called the "ternary" Goldbach conjecture), written in a June 7, 1742 letter to Euler, states "at least it seems that every number that is greater than 2 is the sum of three primes" (Goldbach 1742; Dickson 2005, p. 421). Note that Goldbach considered the...
mathworld.wolfram.com
